I am selling a house but I need to replace the dining room flooring quickly before the house will show well.

I am planning to replace the partially torn vinyl floor in the dining room with carpet. The vinyl material had a hard foam cushioning backing underneath that is in turn installed (cemented I guess) to the floor itself that is concrete.
1. Would you think this is a bad idea in which case I need to go back to the drawing board?

The old vinyl is so hard I cannot cut it with an X-Acto knife or a box cutter or a pocket knife or anything I can think of. The measurer from the carpet store proposed installing the carpet over the existing vinyl after tearing off just the loose partly torn and/or loose portions and using leveling compound.
2. Would you think this is a bad idea in which case I need to go back to the drawing board?

There is a 12 x 12 inch hole in the floor that was jackhammered to go down underneath and repair a broken pipe. It hasn't been patched yet. The measurer suggests patching and leveling all in one move that would bring the patch surface flush with the existing vinyl, not flush with the existing concrete below
3. Would you think this is a bad idea?

If in the future someone wants to change back to vinyl, tearing up everything leaving the patch over the hole between 1/8 and 3/16 of an inch above the original concrete surface.
4. Is it common to put down more leveling compound to ramp up to the patch, or is it desirable to grind down the patch to match the original floor?

I would patch the hole in the floor making the patch level with the rest of the concrete. The elevation for the missing vinyl can be made up with scraps of vinyl flooring, roofing tar paper or anything else you can think of. Then carpet over it.

Slab can not simply be filled with leveling compound. Your slab will be about 4" thick and what I have done in the past is drill holes into he existing slab, inserted short pieces of rebar to tie the patch into the main floor and then filled the hole with concrete. Level as best you can to the existing slab and then use a leveler patch compound or self leveling compound to bring the level up to that of the vinyl. Then pad and carpet over that.